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We are given a pole of height 8 meters, a rope of length 12 meters tied from the top of the pole, and the other end of the rope is fixed to the ground 7 meters away from the base of the pole. The problem asks whether the pole is vertical or inclined.

This problem involves the Pythagorean theorem, which applies to right triangles. The pole, the rope, and the horizontal distance from the base of the pole to where the rope is attached on the ground form a right triangle.

### Using the Pythagorean theorem:
The Pythagorean theorem states that for a right triangle:
\[
a^2 + b^2 = c^2
\]
where:
- \(a\) is the height of the pole (8 meters),
- \(b\) is the horizontal distance from the pole to the point where the rope is fixed on the ground (7 meters),
- \(c\) is the length of the rope (12 meters).

Now, let's check whether the pole is vertical or inclined by calculating whether these values satisfy the Pythagorean theorem:

\[
8^2 + 7^2 = 12^2
\]
\[
64 + 49 = 144
\]
\[
113 \neq 144
\]

Since the two sides of the equation do not match, the pole does **not** form a right triangle with the rope and the ground. This means the pole is **not vertical**; it is inclined.

Would you like to go into more details on this? Here are some related questions you could explore:
1. What would be the required length of the rope if the pole were vertical?
2. How could we calculate the angle of inclination of the pole?
3. What is the significance of the Pythagorean theorem in determining the shape of a structure?
4. How could we use trigonometry to further analyze this problem?
5. If the rope length were different, how would that affect the inclination?

**Tip**: In real-world applications, the Pythagorean theorem is crucial in verifying whether a structure is correctly built or aligned geometrically.

A pole of height 8 m is tightened with a rope of length 12 m at its top and the rope is attached to the ground. The length of the rope's end at the ground to the pole is 7 m. Is the pole built vertically or inclined?

Determine whether a pole of height 8 meters, tied with a 12-meter rope and placed 7 meters from the ground attachment point, is built vertically or inclined. This problem uses the Pythagorean theorem to analyze the geometry of the situation. Suitable for students in Grades 8-10.

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